SOLUTION: What is the smallest possible value of 2x^4 + 8/x^4 over all real nonzero values of x?

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: What is the smallest possible value of 2x^4 + 8/x^4 over all real nonzero values of x?      Log On


   

Question 1027080: What is the smallest possible value of 2x^4 + 8/x^4 over all real nonzero values of x?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Use the derivative,
f%28x%29=2x%5E4%2B8x%5E%28-4%29
df%2Fdx=8x%5E3-32x%5E%28-5%29
Set it equal to zero,
8x%5E3-32x%5E%28-5%29=0
%288x%5E8-32%29%2Fx%5E5=0
8x%5E8=32
x%5E8=1%2F4
x=0+%2B-+%284%29%5E%281%2F8%29
x=0+%2B-+%282%29%5E%281%2F4%29
.
.
.
So then,
f%5Bmax%5D=2%2A%282%5E%281%2F4%29%29%5E4%2B8%2F%28%282%5E%281%2F4%29%29%5E4%29
f%5Bmax%5D=2%2A2%2B8%2F2
f%5Bmax%5D=4%2B4
f%5Bmax%5D=8
.